Supersingular prime

In mathematics, a supersingular prime is a certain kind of prime number.

Formally, let H denote the upper half-plane. For a natural number n, let Γ₀(n) denote the modular group Γ₀, and let w_n be the Fricke involution defined by the block matrix [[0, −1], [n, 0]]. Furthermore, let the modular curve X₀(n) be the compactification (with added cusps) of

Y₀(n) = Γ₀(n)\H,

and for any prime p, define

X₀ ⁺ (p) = X₀(p) / w_p.

Then p is supersingular means by definition that the genus of X₀ ⁺ (p) is zero.

It is also possible to define supersingular primes in a number-theoretic way using supersingular elliptic curves defined over the algebraic closure of the finite field GF(p) that have their j-invariant in GF(p). As it turns out, there are exactly fifteen supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (sequence A002267 in the OEIS). It can also be shown that the supersingular primes are exactly the prime factors of the group order of the Monster group M.

Note the set of supersingular primes is a subset of the set of the Chen primes.